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Sorry but I do not understand the term HTTP API. For me HTTP API is a term related to HTML5 set of Specs designed by the W3C (like Web Socket API, Geolocalization API, ...)

Figure 1.4 is confusing since there is no clear clues that indicate where are the Servers and where is the Client Browser.
This Figure should be redesigned

About the LinkedIn HTTP API example :
Listing 1.3 conveys only Javascript code that is labeled HTTP API. It is quite confusing: if listing 1.3 is about HTTP API, what is the difference between HTTP API and third-party Javascript (listing 1.1)?

My suggestion :
1°) Avoid using the term HTTP API
2°) Figure 1.4 should be the first Figure in this chapter. The left part of this figure should be labeled : the Old Way (doing all processing server-side), the right part of the figure should be labeled : the modern Way (injecting high/low-level functionalities from external server)